STAT4202 Introduction to Mathematical Statistics II

STAT 4202 Introduction to Mathematical Statistics II

[Problem 10.1] If X1, . . . ,Xn constitute a random sample from a population with mean µ, what condition must be imposed on the constants a1, . . . ,an so that

a1X1 + a2X2 + · · · + anXn is an unbiased estimator of µ?

[Problem 10.2] If θ?1 and θ?2 are unbiased estimators of the same parameter θ, what condition must be imposed on the constants k1 and k2 so that

k1θ?1 + k2θ?2

is also an unbiased estimator of θ?

[Problem 10.7] Show that X+1 n+2

is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?

[Problem 10.14] Show that the sample proportion X n

is a minimum variance unbiased esti- mator of the binomial parameter θ. (Hint: Treat X

n as the mean of a random sample of size

n from a Bernoulli population with the parameter θ.)

[Problem 10.16] If Θ?1 and Θ?2 are independent unbiased estimators of a given parameter θ and Var(Θ?1) = 3 · Var(Θ?2), find the constants a1 and a2 such that a1Θ?1 + a2Θ?2 is an unbiased estimator with minimum variance for such a linear combination.

[Problem 10.17] Show that the mean of a random sample of size n from an exponential population is a minimum variance unbiased estimator of the parameter θ.

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Homework 1

[Problem 10.21] If X?1 is the mean of a random sample of size n from a normal population with the mean µ and the variance σ21, X?2 is the mean of a random sample of size n from a normal population with the mean µ and the variance σ22, and the two samples are independent, show that

(a) ω · X?1 + (1 − ω)X?2, where 0 ≤ ω ≤ 1, is an unbiased estimator of µ;

(b) the variance of this estimator is a minimum when

ω = σ22

σ21 + σ 2 2

. (1)

[Problem 10.22] With reference to Exercise 10.21, find the efficiency of the estimator of part (a) with ω = 1

2 relative to this estimator with

ω = σ22

σ21 + σ 2 2

. (2)

[Problem 10.31] Show that if θ? is an estimator of θ, then the MSE of θ? is given by

E[(θ? − θ)2] = V ar(θ?) + [b(θ)]2,

where b(θ) = E(θ?) − θ is the bias of θ? as an estimator for θ.

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